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We propose a measurement system using dual-wavelength digital holography and low-coherence interferometry to measure micro- and nanostructure surface heights. To achieve an extended axial step-measurement range and better image quality, a single light-emitting diode generates two distinct light sources by filtering different center wavelengths and narrower bandwidths. The system can measure surface profile with higher step heights and lower speckle noise in a large field-of-view. Using single-source lighting and a simple configuration, the method supports compactly configured and lower-cost surface-topography measurement systems applicable in various fields. Experimental results for a standard step sample verify the system’s performance.
© 2016 Optical Society of America
1. Introduction
For decades, research in digital holography (DH) has demonstrated its convincing three-dimensional (3D) measurement techniques. DH systems capture images derived from conventional wide-field optical systems. Even so, 3D data depicting objects at nanometer-scale axial resolution with large fields-of-view (FOV) can be achieved only by acquiring several images [1]. In addition, since DH acquires an image from digital image sensors, such as a charge-coupled device (CCD) or a complementary metal-oxide semiconductor, and reconstructs the object information by performing numerical calculations, such as Fresnel diffraction or angular spectrum [2], post-processing methods already used in image or signal processing can be applied. Because of these advantages, DH finds uses in various fields for topographic measurements of samples, such as bio-samples [3], micro-optical components [4], and flat or rough surfaces [5].
For practical applications, however, topographic DH has encountered several drawbacks. First, general DH systems use high-coherent light sources, such as lasers. This guarantees robust and stable operations during measurements, but laser-based interferograms include significant amount of speckle noise, a major source of the degradation of reconstructed object data. Some denoising methods [6] or partial coherence illumination [7] can reduce speckle noise. Even so, the capacity to enhance images remains limited. Second, since DH obtains 3D information from phase information, the measured data are wrapped within the range of [−π π]. The actual value should be calculated with a phase unwrapping process. This process shows severe errors with noisy profiles arising from residual errors [8]. Moreover, the phase-wrap problem defines the measurement limitation of the system in the case of a sample that is stepped or inclined to a degree larger than the half-wavelength difference. Thus, classic DH systems cannot be applied properly to these samples.
To resolve these drawbacks, several methods have been proposed recently. Dual-wavelength DH combines holograms for the same object with different wavelengths, obtaining a larger range of axial heights than does the single-wavelength method [9]. However, using complicated alignments with multiple light sources simultaneously requires multiple sets of holograms to reconstruct the object, and may lead to unexpected errors during hologram acquisition, which creates the need for precise optical components. To reduce the number of the acquisitions, recent works utilized the off-axis [10] or parallel phase-shifting [11] concepts aim to reduce the number of the acquisitions, but the complexity of the systems still remains. Also, to resolve the former DH systems using low-coherence light sources, such as light-emitting diodes (LEDs) have been proposed [12,13]. Although the images obtained from these systems have relatively low speckle noise compared with general laser holograms, the coherence length is also limited, so that the usable interference range is restricted.
In this paper, we propose a dual-wavelength DH system using a single low-coherence light source. Using only one LED with multiple bandpass filters of different center wavelengths, this system can obtain holograms from multiple light sources. Reconstructions obtained with this dual-wavelength holography method permit measurements, at the micrometer scale, of very high stepped surfaces. The proposed system takes advantages of both procedures to achieve reduced speckle noise and a high axial measurement range. Moreover, since it uses only a single light source and less complicated optical alignment, the proposed system can be configured more compactly, yet still offer accuracy similar to the results obtained with conventional surface profilometry systems.
In Section 2, we discuss both dual-wavelength and low-coherence DH, including their principles and limitations. Section 3 depicts and describes the concept and setup of the proposed system. Section 4 presents the results of verifying the proposed system’s performance by measuring a high stepped sample over 1 μm. The result is compared with a probe-based surface profiler, which shows similar accuracy. In Section 5, we summarize the conclusions of the study.
2. Research background
2.1. Dual-wavelength digital holography
The main concept of the dual-wavelength DH system rests on changing the wavelength of the light source on a same object to acquire multiple holograms. In each exposure, the digital image sensor receives the intensity data from the interference pattern between object beam O and reference beam R, which can be expressed as
where λk denotes the wavelength of each light source, and (x, y) are the Cartesian coordinates of the hologram plane. The main goal of the DH reconstruction algorithm is to remove the zero-order term, (|O|2 + |R|2), and the twin image term, (|O*R|), from captured holograms. Several methods, such as in-line [14], off-axis [15], and phase-shifting [16], have been proposed for changing the optical setup, acquisition procedure, and reconstruction algorithm. Also, a Fresnel diffraction-based algorithm can be adapted for propagation to the object plane [17].To obtain topographic measurements of the object using a dual-wavelength scheme, phase data from the complex amplitudes reconstructed in each wavelength are used. With the pairs in the phase profile, the height of the object in a reflective dual-wavelength DH system can be calculated from half of the optical path length (OPL),
where h is the height of the object, is the difference between two phases with different wavelengths and , and Λ is the beat wavelength, which can be calculated fromAs implied in Eq. (3), the beat wavelength has a much larger value than the actual wavelengths used. Usually it increases as the difference between the two wavelengths decreases, ranging from micrometers to millimeters.
Thus, dual-wavelength DH can exceed the axial measurement range, which is limited to half of the wavelength for single-wavelength DH. Because many samples measurable in diffraction-limited lateral resolution have micrometer-scale heights, this system configuration can be applied more widely. However, several disadvantages need to be resolved in order for this system to be useful for accurate measurements. First, since the dual-wavelength setup uses two different light sources in one optical system, it requires more precise alignment than a single-wavelength setup. In addition, because the wavelength varies, distortions or aberrations are generated with different aspects, treating these variations requires using relatively highly accurate optical components. Although utilizing a tunable laser can partially resolve these problems [18], the measurement errors during dual-wavelength reconstruction would be amplified, both by the expanded beat wavelength and by overlapped speckle noises. This leads to using light sources with more stability and less coherence [19]; even so, lasers have limitations for reducing this type of error [20].
2.2. Low-coherence digital holography
A low-coherence DH configuration is analogous to the configuration of the laser-based DH system, except for the light source [21]. In general, a low-coherence light source, such as an LED, is selected for illuminating objects and obtaining holograms. After acquisition, captured interferogram images are reconstructed to calculate complex amplitudes and then propagated to the object plane; this process is equivalent to the usual methodology employed with DH.
Although a low-coherence light source guarantees less speckle noise and better image quality [22], too much low coherence results in poor coherence length, which affects the optical path difference (OPD). To increase the coherence length, both spatial and temporal coherence should be considered. To enhance temporal coherence, a bandpass filter can be adapted for reducing the full-width half-maximum (FWHM) of the wavelength. Since the coherence length is defined as (2ln2/π)(λ2/Δλ), the smaller FWHM of the light source results better temporal coherence. In addition, to increase spatial coherence, collimation using a microscopic lens and spatial filtering with the appropriate radius of aperture are needed. Since the LED source is assumed to be spatially incoherent [22], the increase of speckle noise in filtered beams, compared with coherent imaging systems, is negligible [23]. Note that the coherence length obtained by experiment is shortened due to the spatial coherence properties of the LED [12]. Smaller aperture size can increase the coherence length, however, it also limits the intensity of the initial beam, which is required for appropriate image quality. Therefore, the aperture size might be selected as large enough to obtain appropriate beam power.
After extending the coherence of the light source, the coherence length is in the vicinity of tens of micrometers, which makes possible the implementation of general optical setups for DH, such as those used in off-axis or phase-shifting methods. In an off-axis configuration, because of the occurrence of an OPD in the inclined reference beam, the spatial bandwidth of the off-axis hologram in the LED configuration might be limited. This drawback can be resolved by using a phase-shifting-adapted off-axis setup to achieve acceptable image quality [24].
3. Dual-wavelength digital holography with a single low-coherence light source
As discussed in Section 2, dual-wavelength and low-coherence configurations have been studied in order to overcome the limitations of general DH. Dual-wavelength methods extend the axial range of the step height to be measured. Using a low-coherence light source can reduce speckle noise so that better image quality can be obtained. By combining these two methods, we propose dual-wavelength DH with a single low-coherence light source, which takes advantage of the strengths of both methods.
Figure 1 shows the optical setup of the proposed system. To measure the surface profile, a Michelson interferometer is used. As a low-coherence light source, a single LED with a nominal wavelength of 625 nm and bandwidth of 16 nm (Thorlab M625L3) is selected. To increase coherence, a spatial filter aligned with a collimator and a bandpass filter are adapted for use in front of the system. The collimated and filtered beam is divided by the beamsplitter; the resulting beams then are reflected from the object and the reference mirror, respectively. To capture and reconstruct an image, using the four-step phase-shifting method, a piezoelectric-transducer (PZT) mirror is attached to the reference arm. The interference pattern generated is captured by a CCD that has a 2048 (H) × 2448 (V) array of pixels, where each pixel has dimensions 3.45 μm (H) × 3.45 μm (V) (Sony XCL-C500). To make the system simpler, a lens-free configuration was designed.
Fig. 1 Optical configuration of the proposed system using an LED as single low-coherence light source. The four-step phase-shifting method is implemented for better image quality. LED: light-emitting diode; BF: bandpass filter; OBJ: object; PZT: piezoelectric transducer; CCD: charge-coupled device; and PC: computer.
When the original LED is filtered by the bandpass filter, its FWHM decreases. By using two different filters, having center wavelengths within the bandwidth, two beams with different optical properties can be obtained. In addition, if the difference between the center wavelengths is larger than the bandwidth of the filtered beams, these beams can be distinguished. Thus, by selecting appropriate bandpass filters, the dual-wavelength concept can be adapted to use only one light source. Figure 2 and Table 1 provide the details of a comparison of the measured spectral properties of filtered beams using different bandpass filters. To utilize the beams that are most distinguishable by bandwidth, filters with center wavelengths of 620 nm and 640 nm and bandwidth of 10 nm (Thorlab, FB620-10 and FB640-10, respectively) were selected. As shown in Table 1, the difference of center wavelengths between the two filtered beams (25.97 nm) is much larger than the FWHM of each beam (6.87 nm and 9.17 nm respectively); thus, the two filtered beams can be considered as two distinct light sources. Because the filtered beams are centered close to the border of the original beam, the intensity is reduced to 20% relative to the original. This is a trivial problem for the proposed system, however, because it displays sufficient beam power and image-sensor sensitivity, just as can be found in general commercial products.
Fig. 2 Normalized wavelength distributions of the light sources unfiltered and filtered by bandpass filters.
Table 1. Optical Properties of the Utilized Light Sources
For the four-step phase-shifting configuration, to reconstruct the complex amplitude of the object of two filtered beams, four holograms with modulated references, phased by π/2 per step, are captured in each wavelength. From these images, the object wave at sensor plane can be calculated from Yamaguchi’s method [16]:
where is the phase of the object wave expressed asand is the captured hologram with wavelength λk and reference phase modulated by . While Eq. (4) assumes that the phase profile of the reference wave is uniform at zero, the inhomogeneity of the reference phase occurs in the form of the aberration, so a process for its removal is needed. In addition, since a well-aligned DH system generates an almost uniform or plane-intensity profile in a reference beam, |R| can be ignored in most cases.Using the filtered beams to utilize dual-wavelength DH can extend the maximum step height of the objects. From the center wavelength in Table 1, the beat wavelength is calculated to be 24.2 μm, which is about 40 times larger than the non-filtered light source. Also, compared with the results obtained with lasers having higher coherence, the speckle noise generated from the light source can be greatly reduced, which leads to better measurement quality than what can be obtained with a conventional DH-based system.
4. Experiment results
To verify the performance of the proposed system described in previous section, a standard height sample (VSLI, SHS-1.8QC) was measured. Its surface had patterns with uniform step heights of 1.8 μm. This height is too high for the phase data from the visible-light source to be able to resolve it. Moreover, because the stepped boundary is too steep or almost vertical, even with the help of the phase unwrapping algorithm, the height from the object wave reconstructed from the hologram cannot be calculated. Thus, this object cannot be imaged by a classical single-wavelength DH system.
By using the proposed system denoted in Fig. 1, four-stepped holograms for each wavelength were captured and numerically reconstructed. Figure 3 shows the results obtained for each wavelength. For topographic measurements, the phase images, rather than amplitude, are used. As shown in the cross-section profiles, the phase difference between the step heights does not reflect the actual value because of the phase wrap.
Fig. 3 Holographic reconstruction results using an LED with filters of (a) 620 nm and (b) 640 nm. The cross-section graphs are derived from the red lines in the phase images.
Before attempting reconstruction based on the dual-wavelength concept, additional processes for enhancing the quality of the holograms for each wavelength are applied. In the lensless DH configuration, three major errors occurred during acquisition. First, because of the edge diffraction, unwanted pattern-like noise was generated from the side of the reconstructed image [25]. This error can be minimized by cropping the image and using only the center area. Therefore, the image in Fig. 3 uses a region of 1648 (H) × 1648 (W) pixels for measurements. Second, the included systematic aberration distorts the wavefront of the reconstructed object wave. The source of the aberration lies in the optical components, alignment, or object itself. Third, although a low-coherence DH shows less speckle noise than the image obtained from laser-based interferometry, the remaining noise in each hologram is amplified during dual-wavelength reconstruction because of the extended beat wavelength.
The latter two error types can be resolved with the aberration compensation and dual-wavelength unwrapping methods, respectively. Obtaining the aberration data of the system is possible by capturing and reconstructing the hologram without the object or by approximating the profile directly from the object hologram. After the calculation of the aberrations on the object plane, the numerical lens method can be applied to remove inversely the distortion of the phase [26]. Also, to restrain the amplification of the noise, the phase ambiguity calculated from the dual-wavelength result is applied to the single-wavelength phase map [27,28]:
where and denote the floor and ceiling values respectively. Since Eq. (7) does not use the dual-wavelength reconstructed data directly, the magnitude of the noise follows the single-wavelength profile, which is not amplified. Even though this algorithm requires relatively low phase noise as the beat wavelength is increased, a low-coherence light source satisfies the stated conditions. To reduce speckle noise by temporal coherence, height profiles from each wavelength were averaged [29].Figures 4 presents the measurement results using the proposed system. By using cropping and aberration compensation, the low-frequency degradations included in the reconstructed object waves are completely removed. Applying dual-wavelength unwrapping, as outlined in Eq. (7), results in reduced high-frequency noises. As shown in Figs. 4 (e) - (g), the cross-section profile of the measured sample shows a very flat profile that lacks the disturbances of phase ambiguity and speckle noise.
Fig. 4 Dual-wavelength reconstruction process using the proposed system. Quality-enhanced phase profiles by aberration compensation with (a) 620 nm and (b) 640 nm, respectively. Combined profile (c) using the results from (a) and (b). (d) The area within the red square in (c) enlarged. (e) ~(g) Cross-section profiles of the red lines in (c). The line widths of the regions are (e) 100 µm, (f) 50 µm, and (g) 1 mm, respectively.
From the obtained data, the average height of the sample was measured to be 1.8046 μm and the standard deviation of the flat area was calculated to be 2.63 nm, which was derived with far less noise than when using methods employing a laser-based interferometer. To certify the accuracy of this determination, the same sample was measured with stylus probe profilometer (Bruker DektakXT). Figure 5 shows the comparison between two measurement methods, which present almost identical profiles. By assessing the same region of the sample, the height and standard deviation from the profilometer were measured to be 1.8046 μm and 2.59 nm, which show similar accuracy. Although the probe profilometer has better lateral accuracy, the proposed system can measure up to millimeter-scale large areas with several image acquisitions; this approach is much faster than scanning-based methods.
Fig. 5 (a) Comparison of the measurement data between the result from probe profileometer (Bruker DektakXT) and the proposed system. The compared area is the same region of Fig. 4(g). (b) The zoomed profiles from the dashed square in Fig. 5(a).
Because the proposed system is based on the concept that binds dual-wavelength and low-coherence DH, the configuration for implementation can be varied. In this study, the four-step phase-shifting method was adapted for obtaining the best quality result for the given optical system [30]. Regardless of the degradation of the image and the limitation of the incline angle arising from the short coherence length, the off-axis [10,12,13,24] or parallel phase-shifting [11] method can be adapted to implement a faster and more robust measurement system. Although the lens-less system presented in this paper has a maximized FOV for a given image sensor, additional optical components, such as a microscopic objective lens, can be adapted for providing enhanced lateral resolution. Also, since the proposed system uses a single light source and smaller components, it can be implemented in portable DH applications [31]. Finally, the adaptation of the proposed concept within a transmissive DH system, which will be studied in future research, could prove useful for expanding its applications to obtaining translucent measurement, such as those procured from biological samples, with better accuracy [32].
5. Conclusions
In this paper, we have proposed a DH system for topographic measurement that exhibits higher axial range and image quality. By combining the concepts of a dual-wavelength configuration and low-coherence light source, the proposed system can measure surface profiles with higher steps and less speckle noise than the measurements obtained with single-wavelength laser-based interferometers, despite using the same wavelength light sources. Moreover, since the system uses multiple light sources from a single LED, the system can be smaller and more robust compared with conventional dual-wavelength configurations. Experiments with a standard-height sample have been conducted to validate the accuracy of the proposed system; results show a standard deviation less than 3 nm. These results from the proposed system, when compared with scanning profilometer, demonstrate the potential of the proposed system for use in practical applications. With the advantages of the proposed system, as well as the abilities inherited from conventional DH, these concepts can be deployed in simple and reliable systems in many fields, such as biomedical measurements or product inspection.
Funding
National Research Foundation of Korea (NRF) (2015R1A5A1037668).
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